Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\log _{10} \frac{3}{4}\)

Step-by-Step Solution

Answer

The value of \(\log _{10} \frac{3}{4}\) rounded to three decimal places is -0.125.

Step 1: Understand the Logarithm

Firstly, you need to understand that the given logarithm is base 10. This is written as \(\log _{10}x\). If the base isn't written, it is assumed to be 10. Logarithms are the inverse of exponentiation. It is the power to which a number must be raised to get some other number.

Step 2: Calculate the Logarithm

Now you need to calculate \(\log _{10} \frac{3}{4}\). This can be done with the calculator, that calculating \(\log _{10}\) means finding the exponent to which 10 must be raised to get the value inside the parentheses. In this case, that value is \(\frac{3}{4}\). Most scientific calculators have a log button, which is used for base 10 logarithms.

Step 3: Round the Result

After getting the result, you need to round it to three decimal places. This means you should not consider any digits after the third digit past the decimal point.

Key Concepts

Base 10 Logarithm

The base 10 logarithm, often simply called a common logarithm, is a fundamental concept in mathematics and is denoted as \( \log_{10}(x) \). When the base is not explicitly written, it is usually assumed to be 10. This operation tells us the power to which the base, 10 in this case, must be raised to reach a specific number. For example, \( \log_{10}(100) = 2 \) because 10 raised to the power of 2 equals 100.
Understanding logarithms is essential because they can transform multiplication into addition, which simplifies complex calculations. This property is especially useful in fields like engineering and science, where dealing with very large numbers is common.

• Base 10 is the default unless specified.
• It helps simplify multiplication and division into addition and subtraction.
• It's a vital tool for exponential growth and decay models.

Inverse of Exponentiation

Logarithms serve as the inverse operation to exponentiation. This means that they undo or reverse the process of exponentiating a number. If you have an exponential equation like \( 10^y = x \), the inverse operation using a logarithm would rewrite this as \( y = \log_{10}(x) \).
This relationship is crucial because many problems, especially those involving exponential growth or decay, depend on being able to solve for an exponent. By utilizing logarithms, you can "work backwards" from the result to find what exponent was used, making it easier to handle equations where variable exponents are involved.

• Logarithms reverse exponentiation.
• Useful for solving exponential equations.
• Essential for understanding growth/decay in natural and scientific scenarios.

Scientific Calculators

Scientific calculators are valuable tools for calculating logarithms, especially those of base 10. Most scientific calculators have a dedicated "log" button which allows you to easily input a number and compute its logarithm with base 10. To find \( \log_{10} \left( \frac{3}{4} \right) \), you type \(0.75\) (since \( \frac{3}{4} = 0.75 \)) and press the "log" button.
This great feature of scientific calculators ensures that you can quickly and accurately find logarithmic values without manual calculations. Additionally, these calculators often provide the option to round the result to a certain number of decimal places – a helpful capability when precision is necessary, such as rounding your result to three decimal places.

• They have dedicated log buttons for base 10.
• Allow fast and accurate calculations.
• Include options for rounding to any decimal precision.